Spinning operators and defects in conformal field theory

  • Edoardo Lauria
  • Marco MeineriEmail author
  • Emilio Trevisani
Open Access
Regular Article - Theoretical Physics


We study the kinematics of correlation functions of local and extended operators in a conformal field theory. We present a new method for constructing the tensor structures associated to primary operators in an arbitrary bosonic representation of the Lorentz group. The recipe yields the explicit structures in embedding space, and can be applied to any correlator of local operators, with or without a defect. We then focus on the two-point function of traceless symmetric primaries in the presence of a conformal defect, and explain how to compute the conformal blocks. In particular, we illustrate various techniques to generate the bulk channel blocks either from a radial expansion or by acting with differential operators on simpler seed blocks. For the defect channel, we detail a method to compute the blocks in closed form, in terms of projectors into mixed symmetry representations of the orthogonal group.


Boundary Quantum Field Theory Conformal Field Theory Space-Time Symmetries Wilson, ’t Hooft and Polyakov loops 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited


  1. [1]
    J.L. Cardy, Conformal invariance and surface critical behavior, Nucl. Phys. B 240 (1984) 514 [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    P. Liendo, L. Rastelli and B.C. van Rees, The bootstrap program for boundary CFT d, JHEP 07 (2013) 113 [arXiv:1210.4258] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    D. Gaiotto, D. Mazac and M.F. Paulos, Bootstrapping the 3d Ising twist defect, JHEP 03 (2014)100 [arXiv:1310.5078] [INSPIRE].
  4. [4]
    F. Gliozzi, P. Liendo, M. Meineri and A. Rago, Boundary and interface CFTs from the conformal bootstrap, JHEP 05 (2015) 036 [arXiv:1502.07217] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    F. Gliozzi, Truncatable bootstrap equations in algebraic form and critical surface exponents, JHEP 10 (2016) 037 [arXiv:1605.04175] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    P. Liendo, C. Meneghelli and V. Mitev, Bootstrapping the half-BPS line defect, JHEP 10 (2018) 077 [arXiv:1806.01862] [INSPIRE].
  7. [7]
    M. Hogervorst, Crossing kernels for boundary and crosscap CFTs, arXiv:1703.08159 [INSPIRE].
  8. [8]
    M. Lemos, P. Liendo, M. Meineri and S. Sarkar, Universality at large transverse spin in defect CFT, JHEP 09 (2018) 091 [arXiv:1712.08185] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    D. Poland, S. Rychkov and A. Vichi, The conformal bootstrap: theory, numerical techniques and applications, Rev. Mod. Phys. 91 (2019) 015002 [arXiv:1805.04405] [INSPIRE].
  10. [10]
    A. Dymarsky, J. Penedones, E. Trevisani and A. Vichi, Charting the space of 3D CFTs with a continuous global symmetry, JHEP 05 (2019) 098 [arXiv:1705.04278] [INSPIRE].ADSGoogle Scholar
  11. [11]
    A. Dymarsky, On the four-point function of the stress-energy tensors in a CFT, JHEP 10 (2015) 075 [arXiv:1311.4546] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    A. Dymarsky, F. Kos, P. Kravchuk, D. Poland and D. Simmons-Duffin, The 3d stress-tensor bootstrap, JHEP 02 (2018) 164 [arXiv:1708.05718] [INSPIRE].
  13. [13]
    S. Balakrishnan, T. Faulkner, Z.U. Khandker and H. Wang, A general proof of the quantum null energy condition, arXiv:1706.09432 [INSPIRE].
  14. [14]
    L.-Y. Hung, R.C. Myers and M. Smolkin, Twist operators in higher dimensions, JHEP 10 (2014) 178 [arXiv:1407.6429] [INSPIRE].
  15. [15]
    L. Bianchi, M. Meineri, R.C. Myers and M. Smolkin, Rényi entropy and conformal defects, JHEP 07 (2016) 076 [arXiv:1511.06713] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  16. [16]
    A. Lewkowycz and J. Maldacena, Exact results for the entanglement entropy and the energy radiated by a quark, JHEP 05 (2014) 025 [arXiv:1312.5682] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  17. [17]
    B. Fiol, E. Gerchkovitz and Z. Komargodski, Exact bremsstrahlung function in N = 2 superconformal field theories, Phys. Rev. Lett. 116 (2016) 081601 [arXiv:1510.01332] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    L. Bianchi, M. Lemos and M. Meineri, Line defects and radiation in N = 2 conformal theories, Phys. Rev. Lett. 121 (2018) 141601 [arXiv:1805.04111] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    D.M. McAvity and H. Osborn, Conformal field theories near a boundary in general dimensions, Nucl. Phys. B 455 (1995) 522 [cond-mat/9505127] [INSPIRE].
  20. [20]
    M. Billò, V. Gonçalves, E. Lauria and M. Meineri, Defects in conformal field theory, JHEP 04 (2016) 091 [arXiv:1601.02883] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  21. [21]
    S. Guha and B. Nagaraj, Correlators of mixed symmetry operators in defect CFTs, JHEP 10 (2018) 198 [arXiv:1805.12341] [INSPIRE].
  22. [22]
    M. Fukuda, N. Kobayashi and T. Nishioka, Operator product expansion for conformal defects, JHEP 01 (2018) 013 [arXiv:1710.11165] [INSPIRE].
  23. [23]
    P. Liendo and C. Meneghelli, Bootstrap equations for N = 4 SYM with defects, JHEP 01 (2017) 122 [arXiv:1608.05126] [INSPIRE].
  24. [24]
    L. Rastelli and X. Zhou, The Mellin formalism for boundary CFT d , JHEP 10 (2017) 146 [arXiv:1705.05362] [INSPIRE].
  25. [25]
    V. Goncalves and G. Itsios, A note on defect Mellin amplitudes, arXiv:1803.06721 [INSPIRE].
  26. [26]
    N. Kobayashi and T. Nishioka, Spinning conformal defects, JHEP 09 (2018) 134 [arXiv:1805.05967] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    M. Billò, M. Caselle, D. Gaiotto, F. Gliozzi, M. Meineri and R. Pellegrini, Line defects in the 3d Ising model, JHEP 07 (2013) 055 [arXiv:1304.4110] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    E. Lauria, M. Meineri and E. Trevisani, Radial coordinates for defect CFTs, JHEP 11 (2018) 148 [arXiv:1712.07668] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    M. Isachenkov, P. Liendo, Y. Linke and V. Schomerus, Calogero-Sutherland approach to defect blocks, JHEP 10 (2018) 204 [arXiv:1806.09703] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    A. Gadde, Conformal constraints on defects, arXiv:1602.06354 [INSPIRE].
  31. [31]
    M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal blocks, JHEP 11 (2011) 154 [arXiv:1109.6321] [INSPIRE].
  32. [32]
    M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal correlators, JHEP 11 (2011) 071 [arXiv:1107.3554] [INSPIRE].
  33. [33]
    M.S. Costa and T. Hansen, Conformal correlators of mixed-symmetry tensors, JHEP 02 (2015) 151 [arXiv:1411.7351] [INSPIRE].
  34. [34]
    M.S. Costa, T. Hansen, J. Penedones and E. Trevisani, Projectors and seed conformal blocks for traceless mixed-symmetry tensors, JHEP 07 (2016) 018 [arXiv:1603.05551] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    V.K. Dobrev, V.B. Petkova, S.G. Petrova and I.T. Todorov, Dynamical derivation of vacuum operator product expansion in Euclidean conformal quantum field theory, Phys. Rev. D 13 (1976)887 [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    F. Rejon-Barrera and D. Robbins, Scalar-vector bootstrap, JHEP 01 (2016) 139 [arXiv:1508.02676] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    M.S. Costa and T. Hansen, AdS weight shifting operators, JHEP 09 (2018) 040 [arXiv:1805.01492] [INSPIRE].
  38. [38]
    D. Karateev, P. Kravchuk and D. Simmons-Duffin, Weight shifting operators and conformal blocks, JHEP 02 (2018) 081 [arXiv:1706.07813] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal blocks, JHEP 11 (2011) 154 [arXiv:1109.6321] [INSPIRE].
  40. [40]
    A. Castedo Echeverri, E. Elkhidir, D. Karateev and M. Serone, Deconstructing conformal blocks in 4D CFT, JHEP 08 (2015) 101 [arXiv:1505.03750] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    P. Kravchuk and D. Simmons-Duffin, Counting conformal correlators, JHEP 02 (2018) 096 [arXiv:1612.08987] [INSPIRE].
  42. [42]
    A.B. Zamolodchikov, Conformal symmetry in two-dimensions: an explicit recurrence formula for the conformal partial wave amplitude, Commun. Math. Phys. 96 (1984) 419 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  43. [43]
    M. Hogervorst and S. Rychkov, Radial coordinates for conformal blocks, Phys. Rev. D 87 (2013) 106004 [arXiv:1303.1111] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    M.S. Costa, T. Hansen, J. Penedones and E. Trevisani, Radial expansion for spinning conformal blocks, JHEP 07 (2016) 057 [arXiv:1603.05552] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping mixed correlators in the 3D Ising model, JHEP 11 (2014) 109 [arXiv:1406.4858] [INSPIRE].
  46. [46]
    F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping the O(N) vector models, JHEP 06 (2014) 091 [arXiv:1307.6856] [INSPIRE].
  47. [47]
    J. Penedones, E. Trevisani and M. Yamazaki, Recursion relations for conformal blocks, JHEP 09 (2016) 070 [arXiv:1509.00428] [INSPIRE].
  48. [48]
    P. Kravchuk and D. Simmons-Duffin, Light-ray operators in conformal field theory, JHEP 11 (2018) 102 [arXiv:1805.00098] [INSPIRE].
  49. [49]
    F.A. Dolan, Character formulae and partition functions in higher dimensional conformal field theory, J. Math. Phys. 47 (2006) 062303 [hep-th/0508031] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    R. de Mello Koch, P. Rabambi, R. Rabe and S. Ramgoolam, Counting and construction of holomorphic primary fields in free CFT4 from rings of functions on Calabi-Yau orbifolds, JHEP 08 (2017) 077 [arXiv:1705.06702] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  51. [51]
    H. Osborn and A.C. Petkou, Implications of conformal invariance in field theories for general dimensions, Annals Phys. 231 (1994) 311 [hep-th/9307010] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  52. [52]
    L. Bianchi, M. Preti and E. Vescovi, Exact bremsstrahlung functions in ABJM theory, JHEP 07 (2018) 060 [arXiv:1802.07726] [INSPIRE].ADSMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  • Edoardo Lauria
    • 1
  • Marco Meineri
    • 2
    Email author
  • Emilio Trevisani
    • 3
    • 4
  1. 1.Instituut voor Theoretische Fysica, KU LeuvenLeuvenBelgium
  2. 2.Institute of Physics, É cole Polytechnique Fédérale de Lausanne (EPFL)LausanneSwitzerland
  3. 3.Laboratoire de Physique ThéoriqueÉcole Normale Supérieure & PSL Research UniversityParis Cedex 05France
  4. 4.Institut des Hautes Études ScientifiquesBures-sur-YvetteFrance

Personalised recommendations